A wire of length `L` is shaped into a circle and then folded as shown. Magnetic moment of the frame when a current `i` is passed through it is:

Options

A. `(L^(2)i)/(4pi)`
B. `(L^(2)i)/(4sqrt2pi)`
C. `(L^(2)i)/(4)`
D. `(L^(2)i)/(8pi)`

Answer

B

Explanation

Since `L=2pir, r=(L)/(2pi)`
Area of circle `pir^(2)=(piL^(2))/(4pi^(2))=(L^(2))/(4pi)`
Area of each half of circle `=(L^(2))/(8pi)`
Magnetic moment of each half `=(L^(2)i)/(8pi)`
The two magnetic moments are at right angles to each other.
Magnetic moment of the frame
`=sqrt(((L^(2)i)/(8pi))^(2)+((L^(2)i)/(8pi))^(2))=(sqrt2.L^(2)i)/(8pi)=(L^(2)i)/(4sqrt(2pi))`